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1    NORMED ALGEBRAS       307
Examples of Narmed Algebras

(15.1.4)    For each nonempty set X, the set ^C(X) of bounded complex-valued
functions on X is a commutative Banach algebra with respect to the norm
(7.1.1) and ordinary multiplication of functions: the inequality (15.1.1) is
clearly satisfied, the constant function 1 is the unit element of ^CPO and
satisfies (15.1.2). If X is a topological space the subspace ^c(X) of bounded
continuous functions on X is a closed subalgebra of ^C(X). If X is metrizable,
separable and locally compact, the space ^(X) (13.20.5) of continuous func-
tions which tend to 0 at infinity, and the space Jf* C(X) of continuous functions
with compact support, are ideals in the algebra ^(X), the former being the
closure of the latter.

(15.1.5)    If X is the closed disk \z\ g 1 in C, then the set j/(X) c <jpc(X) of
continuous functions on X which are analytic in the interior \z\ < 1 of X is a
closed subalgebra of #C(X), by virtue of the theorem of convergence of analytic
functions (9.12.1).

(15.1.6)    We have already seen (11.1) that the algebra J?(E) of continuous
endomorphisms of a complex normed space E is a normed algebra (in general
noncommutative) with the identity mapping 1E of E as unit element (the
inequality (15.1.1) in this case is just (5.7.5), and the relation (15.1.2) is
obvious). Furthermore, if E is a Banach space, then JSf(E) is a Banach algebra
(5.7.3).

The set of compact operators in J?(E) is a closed two-sided ideal by virtue
of (11.2.6) and (11.2.10).

(15.1.7)    Let G be a separable metrizable locally compact group. The set
M(G) of bounded measures on G, endowed with the convolution product
(A, ju) I-H> A * n an<i the norm (13.20.1) is a Banach algebra. For in this case the
inequality (15.1.1) is just (14.6.2.1) with n = 2; the unit element se (the Dirac
measure at the neutral element e of G) is such that ||ee|| = 1; and Mc(G), being
the dual of the normed space *c(G) (13.20.6), is complete (5.7.3). The sub-
space of Mc(G) consisting of measures with base a left Haar measure /? on G
can be identified (together with its norm) with the Banach space L(G, /?),
because Nx(/) = ||/- ]B|| (13.20.3), and it follows from (14.9.2) that if G is
unimodular, L(G, /?) is a closed two-sided ideal in M(G). We shall always
identify L(G, /?) with this ideal (the identification of course depends on the
choice of/?). Under this identification, Jf C(G) is identified with a subalgebra
(14.10.5) of Mc(G), which is not, in general, an ideal (unless G is compact
(14.9.1)) and, in general, is not closed, because its closure is Lc(G, /}) (13.11.6)..1.1)                                        \\xy\\ ^ \\x\\  \\y\\